How to apply a displacement modifier to one area of an object from an image?
I think you'll need to use a UV map here as using a vertex group does not seem to give a good result.

As you want to use a displace modifier you'll need to subdivide your shape a lot. But, whatever the subdivision level is, the precision of the displacement will be limited by the texture resolution itself.
So, if the model is to be 3D printed, both resolutions (subdivisions and texture) will matter.

First, make your UV map :


Select your face in edit mode
Go to front view (or more generally in front of the face)
UV unwrap : U then choose "project from view (bounds)"





Adjust your UV map 




Note : you may also UV unwrap all the faces and / or make your texture after the unwrapping (so, or adjust the UV geometry to the texture, or make a texture adapted to the UV geometry)


Set your texture as displacement
And use UV as texture coordinates





Subdivide your face (a lot)





Adjust the displacement




Note 

If you need to do this on several faces :


UV unwrap each face and eventually use the vertex group option of the displace modifier (a vertex group per face, but taking care of the border vertices). Alternative : use several UV maps.
I also suggest to apply the subdivision to all faces at the same time.


Again, I am not sure that the precision will be good enough for a 3D printed object.